The Stacks project

Remark 85.15.7 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/U}$. Then we can combine the constructions given in Remarks 85.15.5 and 85.15.6 to get

  1. a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C}, X)$,

  2. a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,

  3. a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$ of ringed topoi,

  4. a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.

Of course all of the results mentioned in Remark 85.15.6 hold in this setting as well.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0D9V. Beware of the difference between the letter 'O' and the digit '0'.